 Solution

ID: past-exam-of-the-real-and-harmonic-analysis-course-of-the-university-of-oxford/2007/2/c/solution

(X, d)

is an outer measure satisfying

d (A, B) > 0 ⟹ μ (A \cup B) = μ (A) + μ (B) .

Here

d (A, B) = in f {d (a, b) : a \in A, b \in B}

. Every Borel set is measurable for a metric outer measure. Consequently its measurable sigma-algebra contains the Borel sigma-algebra; it may be strictly larger.

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